Simulation of Prestressed Steel Fiber Concrete Beams Subjectedto Shear

Liang Lu1),*, P. R. Tadepalli2), Y. L. Mo3), and T. T. C. Hsu3)

(Received November 13, 2015, Accepted May 22, 2016, Published online June 16, 2016)

Abstract: This paper developed an analytical software, called Simulation of Concrete Structures (SCS), which is used for

numerical analysis of shear-critical prestressed steel fiber concrete structures. Based on the previous research at the University of

Houston (UH), SCS has been derived from an object-oriented software framework called Open System for Earthquake Engi-

neering Simulation (OpenSees). OpenSees was originally developed at the University of California, Berkeley. New module has

been created for steel fiber concrete under prestress based on the constitutive relationships of this material developed at UH. This

new material module has been integrated with the existing material modules in OpenSees. SCS thus developed has been used for

predicting the behavior of the prestressed steel fiber concrete I-beams and Box-beams tested earlier in this research. The analysis

could well predict the entire behavior of the beams including the elastic stiffness, yield point, post-yield stiffness, and maximum

load for both web shear and flexure shear failure modes.

Keywords: simulation of concrete structure, prestressed steel fiber concrete, OpenSees, constitutive relationship, shear.

1. Introduction

OpenSees is an object-oriented computer program inseismic engineering (Fenves 2015). The features of Open-Sees are described on the OpenSees website (www.opensees.berkeley.edu) and presented briefly here.The architecture of OpenSees consists of four objects:

ModelBuilder, Domain, Analysis, and Recorder under theOpenSees framework, as shown in Fig. 1.The ModelBuilder object constructs the nodes and masses

on the nodes, creates the elements and materials of the ele-ments, defines the loads exerting on the nodes and the ele-ments, and defines the constraints exerting on the nodes. TheModelBuilder is responsible for building the objects in themodel such as Node, Mass, Material, Element, LoadPattern,Constraint, etc. and adding them to the Domain object.After the objects are created by the ModelBuilder object,

they are stored in the Domain object. The Domain objectalso provides access of Analysis and Recorder objects to theobjects in the Domain and holds the state of the modelduring the analysis procedure.

The Analysis object is responsible for performing static ordynamic analysis on the model (Fenves 2015). The OpenSeesis not able to perform nonlinear analysis on membrane struc-tures such as reinforced concrete panels and shear wallsbecause there is no membrane model in it. Also, the uniaxialmaterial models of steel and concrete in OpenSees are toosimple, and more sophisticated models need to be created. Forexample, the concrete model Concrete01 in OpenSees doesnot consider the stress of concrete in tension and the softeningeffect in compression due to the biaxial stress status. The steelmodel Steel01 does not consider the Bauschinger effect in thereloading and unloading paths. Therefore, the new finite ele-ment models of reinforced concrete material could be installedinto OpenSees for performing analysis on reinforced concretemembrane structures.

2. Softened Membrane Modelfor Prestressed Steel Fiber Concrete (SMM-

PSFC)

SMM-PSFC (Hoffman 2010)was developed to simulate theentire behavior of PSFC elements under monotonic loading.SMM-PSFC consisted of the stress equilibrium and straincompatibility equations along with the constitutive models ofmaterials. However, SMM-PSFC contains the uniaxial con-stitutive models of materials under monotonic loading only.

2.1 Stress Equilibrium and Strain CompatibilityEquationsFigure 2 shows an in-plane element model. The element

considered is reinforced with two orthogonal grids of

1)Research Institute of Structural Engineering and

Disaster Reduction, Tongji University, Shanghai 200092,

China.

*Corresponding Author; E-mail: 95010@tongji.edu.cn2)American Global Maritime, Houston, TX 77079, USA.3)Department of Civil and Environmental Engineering,

University of Houston, Houston, TX 77204, USA.

Copyright � The Author(s) 2016. This article is published

with open access at Springerlink.com

International Journal of Concrete Structures and MaterialsVol.10, No.3, pp.297–306, September 2016DOI 10.1007/s40069-016-0153-8ISSN 1976-0485 / eISSN 2234-1315

297

prestressing tendons and mild steel bars. The first Cartesian‘� t coordinate system is along the steel bar directions. Thesecond Cartesian 1–2 coordinates is along the principalstress. For the analytical purposes, it is assumed that themembrane element thickness is uniform with the steel barsare uniformly distributed in two orthogonal directions. Thefour applied stresses exerting on the element edges areassumed to be uniformly distributed.In a membrane element, the external applied stresses (r‘,rt

and s‘t) can be indicated by the prestressing steel stresses (f‘pand ftp), the mild steel stresses (f‘ and ft), and the internalstresses of concrete (rc2,r

c1 and sc21), the three stress equi-

librium equations are given in Eqs. (1) to (3).

r‘ ¼ rc2 cos2 a2 þ rc1 sin

2 a2 þ sc212 sin a2 cos a2 þ q‘f‘þ q‘pf‘p

ð1Þ

rt ¼ rc2 sin2 a2 þ rc1 cos

2 a2 � sc212 sin a2 cos a2 þ qt ftþ qtpftp;

ð2Þ

s‘t ¼ ð�rc2 þ rc1Þ sin a2 cos a2 þ sc21ðcos2 a2 � sin2 a2Þð3Þ

The strains (e1, e2 and c21) in 1–2 coordinates can beconverted to the strains (e‘, et and c‘t), as shown in Eqs. (4)to (6) (Pang and Hsu 1996).

e‘ ¼ e2 cos2 a2 þ e1 sin

2 a2 þc212

2 sin a2 cos a2 ð4Þ

et ¼ e2 sin2 a2 þ e1 cos

2 a2 �c212

2 sin a2 cos a2 ð5Þ

c‘t2

¼ ð�e2 þ e1Þ sin a2 cos a2 þc212

ðcos2 a2 � sin2 a2Þ

ð6Þ

2.2 Biaxial Strains Convert to Uniaxial StrainsSince general lab experiments and reference literatures can

give only the uniaxial constitutive laws of steel and concrete,only the uniaxial constitutive laws can be utilized by ageneral analytical software, the biaxial strains in aboveequations should be transformed to uniaxial strains. Thus,using the Hsu/Zhu ratios (m12,m21), four equations has beenderived (Zhu and Hsu 2002) to represent the relationshipbetween the uniaxial strains (e1; e2; el and et) and the biaxialstrains (e1,e2,e‘ and et), as shown in Eqs. (7) to (10).

�e1 ¼1

1� m12m21e1 þ

m121� m12m21

e2 ð7Þ

�e2 ¼m21

1� m12m21e1 þ

1

1� m12m21e2 ð8Þ

�e‘ ¼ �e2 cos2 a2 þ �e1 sin

2 a2 þc122

2 sin a2 cos a2 ð9Þ

�et ¼ �e2 sin2 a2 þ �e1 cos

2 a2 �c122

2 sin a2 cos a2 ð10Þ

The uniaxial strains e1; e2; el and et can be calculated byEqs. (7) to (10), then the stresses rc1, r

c2, s

c12, f‘ and ft in

Eqs. (1) to (3) can be obtained using the uniaxial constitutivelaws.Under monotonic shear stresses, two Hsu/Zhu ratios of

panels can be given in Eqs. (11) and (12) (Zhu and Hsu2002).

m12 ¼0:2þ 850esf ; esf � ey

1:9; esf [ ey

�ð11Þ

Fig. 1 Principal objects in OpenSees (Fenves 2015).

Fig. 2 A typical reinforced concrete plane stress element.

298 | International Journal of Concrete Structures and Materials (Vol.10, No.3, September 2016)

m21 ¼ 0 ð12Þ

Where, esf is the average (smeared) tensile strain of steelrebar in the ‘� or the t� direction, whichever yields first,taken to calculate the Hsu/Zhu ratio m12.

2.3 Constitutive Laws of Materials2.3.1 Uniaxial Constitutive Laws of Prestressed

Steel Fiber ConcreteThe constitutive model for PSFC along with the factors

that will affect PSFC are summarized in this section. Theresults are plotted in Fig. 3. Note that the tensile stress isapplied in 1-direction and the compressive stress in 2-di-rection. Development of these constitutive relationships hasbeen reported by Hoffman (2010). These proposed consti-tutive laws of PSFC takes consideration on the effect ofpresence of the steel fibers in the concrete.

2.3.2 SFC in TensionThe relationships of rc1 and the uniaxial strain e1 of pre-

stressed SFC are given as follows:Stage UC:

rc1 ¼ E0c�e1 þ rci;�e1 � �ecx � �ecið Þ ð13aÞ

Stage T1:

rc1 ¼ E00c ð�e1 þ �eciÞ; �ecx � �ecið Þ\�e1 � ecy � �eci

� �ð13bÞ

Stage T2:

rc1 ¼ E000c ð�e1 þ �eciÞ; �ecy � �eci

� �\�e1 � ecult � �ecið Þ ð13cÞ

Stage T3:

rc1 ¼ EIVc ð�e1 þ �eciÞ;�e1 [ ecult � �ecið Þ ð13dÞ

Where,E0c = decompression modulus of concrete given as 2f 0c

e0rci = initial stress in SFC�eci = initial strain in concrete due to prestress�ecx = extra concrete strain after decompression, taken as

�eci � rciE0c

ecmax = SFC maximum strain calculated by 0.04-epi,where, epi = initial uniaxial strain of prestressing tendonsecult = SFC strain under ultimate stress, calculated by

0.01—epifcult = SFC ultimate stress, taken as ð0:2FF þ 12qlÞ

ffiffiffiffif 0c

p,

where, FF = fiber factor, ql = longitudinal steel ratioecy = SFC yield strain taken as 0.0005fcy = SFC effective yield stress, taken as

0:4 � FF � CFffiffiffiffif 0c

p, (f 0c and

ffiffiffiffif 0c

pare in MPa), CF = 1 for

SFC tensile volume confined (sandwiched) by two or moretendons, or CF = � for SFC tensile volume unconfined bytendonsE00c = modulus of SFC, taken as

fcyecy��ecx

E000c = modulus of SFC, taken as

fcult�fcyecult�ecy

EIVc = modulus of SFC, taken as �fcult

emax�ecult

2.3.3 SFC in CompressionThe average (smeared) constitutive laws of SFC com-

pression stress rc2 and the uniaxial compression strain �e2 aretaken as follows:

rc2 ¼ ff 0c 2�e2fe0

� �� �e2

fe0

� �2" #

;�e2fe0

� 1; ð14aÞ

or

Fig. 3 Average stress–strain relationships of prestressed steel fiber concrete.

International Journal of Concrete Structures and Materials (Vol.10, No.3, September 2016) | 299

rc2 ¼ ff 0c 1� �e2=fe0 � 1

4=f� 1

� �2" #

;�e2fe0

[ 1 ð14bÞ

Where, f is the softening coefficient, which can be cal-culated as follows:

f ¼ f f 0c� �

f �e1ð Þf bð ÞWpWf � 0:9; ð15Þ

Where,

f f 0c� �

¼ 5:8ffiffiffiffif 0c

p � 0:9ðf 0c inMPaÞ; ð16Þ

f �e1ð Þ ¼ 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 400�e1

p ; ð17Þ

f bð Þ ¼ 1� bj j24�

; b ¼ 1

2tan�1 c21

e2 � e1ð Þ

� ; ð18Þ

Wp ¼ 1:15þ bj j 0:09 bj j � 1ð Þ6

; ð19Þ

and

Wf ¼ 1þ 0:2FF ð20Þ

2.3.4 SFC in ShearIn the 1–2 coordinates, the relationship between the con-

crete stress (sc12) and the strain (c12) is reported by Zhu et al.(2001) and shown in Eq. (21).

sc12 ¼rc1 � rc22ðe1 � e2Þ

c12 ð21Þ

where rc1 and rc2 are the average (smeared) concrete stresses;e1 and e2 are the biaxial smeared strains in the 1- and 2-directions of the principal applied stresses, respectively.

2.3.5 Prestressing Tendons Embedded in SFCThe prestressing tendons are embedded in SFC. The

average (smeared) stress–strain relationship of PSFC isgiven as follows:

fps ¼ Eps�e0ses\

0:7fpuEps

; ð22aÞ

or

fps ¼E00ps�e

0s

1þ E00ps�e

0s

f 0pu

�5� 1

5

;�e0s �0:7fpuEps

; ð22bÞ

Where,Eps = elastic modulus of prestressing strands, 200GPa�e0s = �es ? edec, uniaxial steel bar strainfpu = ultimate strength of prestressing strands, 1862 MPa

E00ps = modulus of prestressing strands in plastic area

(Eq. (22)), 209 GPaf 0pu = modified strength of prestressing strands, 1793 MPaIn the above equations, ps can be exchanged by ‘p and tp

for the longitudinal tendons and the transverse tendonsrespectively (Fig. 4).

2.3.6 Mild Steel Embedded in SFCThe mild steel bars are installed in concrete as those in

SMM. The average (smeared) stress–strain relationships canbe expressed as follows:Stage 1:

fs ¼ Es�es;�es ��en ð23Þ

Stage 2:

fs ¼ fy ��ð1� 0:096FFÞð0:91� 2BÞ þ ð0:2FF þ 1Þ:

�ð0:02þ 0:25BÞ�e0s

ey

;�es [�en

ð24Þ

Stage 3 (unloading):

fs ¼ fp � Esð�ep � �esÞ;�es\�ep ð25Þ

Where,

�en ¼ eyð0:93� 2BÞ ð26aÞ

B ¼ 1

qfcrfy

� �1:5

ð26bÞ

ecr = 0.00008, concrete cracking strain

Fig. 4 Constitutive relationship of prestressing tendons.

300 | International Journal of Concrete Structures and Materials (Vol.10, No.3, September 2016)

fcr = 0:31ffiffiffiffif 0c

p(f 0c and

ffiffiffiffif 0c

pare in MPa), concrete cracking

stress (Fig. 5)

2.4 Experimental VerificationThe SMM-PSFC was used to realize the experimental

shear behavior of PSFC membrane elements with differentsteel grid orientations and steel percentages (Laskar et al.2014). Figure 6 shows the validity of SMM-PSFC in pre-dicting the behavior of PSFC panels under pure shear. Thefigure shows the comparison between the analytical and themeasured shear stress–strain curves of a PSFC panel byHoffman (2010) with results predicted by SMM-PSFC. Itcan be seen that the predictions from SMM-PSFC are ingood agreement.

3. Simulation of PSFC Beams

The PSFC beams shown in Fig. 7 have been analysedusing SCS. Details and shear behavioral characteristics of

the beams are given in Tadepalli et al. (2014, 2015). Sometests of PSFC beams were carried out to validate the SCSprogram, which was developed using the constitutive laws ofPSFC in SMM-PSFC (Hoffman 2010). Since the PSFCbeams were tested under monotonic loads, this validationand applicability of SCS program is only suitable in pre-dicting the behaviour of PSFC structures under monotonicloading.

3.1 Analytical ModelTwo types of elements were chosen to analyze the PSFC

beams. The prestressing loads exerting on the beam weretreated as external forces. These loads are the total pre-stressing forces after losses. The top and bottom flanges ofthe PSFC beams were represented by Nonlin-earBeamColumn elements and the web of the beams weremodeled as PCPlaneStress quadrilateral elements.All the concentrated loads applied on the beam in the

model acted at nodes. The effect of bearing plates, whichwere actually used in the load-test to apply vertical loads onthe beam, was ignored for simplicity. The loads were dis-tributed among three nodes adjacent to the location of theapplied load. The analysis yielded similar results in the caseswhen (a) larger load was applied at the node correspondingto the actual loading point and lower loads were applied atthe two adjacent nodes and (b) loads were distributedequally among the three nodes. Hence, the concentratedvertical loads on the beams were modeled at a node corre-sponding to the actual loading point and two adjacent nodes.The results of beam analysis i.e. vertical forces and nodaldeflections, were computed at every loading step that hadnumerically converged. Additionally, the stresses and strainsin the beam elements were also calculated by the program(Laskar 2009).

3.2 SCS FEM Models of PSFC Beams3.2.1 I-BeamsThe PSFC I-beams was meshed using the SCS program, as

represented in Figs. 8 and 9. Sixteen and fifteen Nonlin-earBeamColumn elements were used to simulate the bottomand top flanges in the case of web-shear and flexure-shearfailure modes, respectively. The method of the cross sectiondiscretization of the two flanges is that forty equivalentrectangular NonlinearBeamColumn elements are meshed tomodel each flange, as shown in Fig. 10. Two of the twelvetendons in the specimens were provided in the Nonlin-earBeamColumn elements representing the bottom flange ofthe specimens. The remaining tendons were provided in thequadrilateral elements used to represent the webs of thesespecimens.The tendons have initial strains, it was taken intoaccount by using the TendonL01 material module. The steeland concrete fibers were defined by using Steel02 andConcrete01 constitutive modules, respectively.The Concrete01 module used herein was a uniaxial

material module of concrete previously created in OpenSeesfollowing the modified Kent and Park model (Park et al.1982). The steel02 module used in the SCS was a uniaxialMenegotto-Pinto object (Dhakal and Maekawa 2002). This

Fig. 5 Stress-strain relationship of reinforcing steel usingtwo-straight line expression.

Fig. 6 Predicted versus experimental shear stress–straincurves of a typical PSFC panel (Hoffman 2010).

International Journal of Concrete Structures and Materials (Vol.10, No.3, September 2016) | 301

Fig. 7 Cross sections of I-beam and box-beam.

Fig. 8 FEM model of PSFC I-beams tested under web-shear.

Fig. 9 FEM model of PSFC I-beams tested under flexural-shear.

Fig. 10 Cross-section discretization for PSFC I-beams governed by web-shear.

302 | International Journal of Concrete Structures and Materials (Vol.10, No.3, September 2016)

material object also allowed user to enter the initial strain inthe steel, which is useful in modeling the prestressingstrands.The web of the beams have been divided into sixteen and

fifteen PCPlaneStress quadrilateral elements in case of webshear and flexural shear, respectively. The steel ratio in the

vertical direction was taken as a very small number to avoidnumerical problems during the analysis.

3.2.2 Box-BeamsThe PSFC Box-beams was meshed using the SCS pro-

gram, as shown in Figs. 11, 12 and 13. The two flanges in

Fig. 11 FEM model of PSFC box-beams tested under web-shear (a/d = 1.8).

Fig. 12 FEM model of box-beams tested under web-shear (a/d = 2.5).

Fig. 13 FEM model of box-beams tested under flexure-shear (a/d = 4.1).

Fig. 14 Cross-section discretization for PSFC box-beams governed by web-shear.

International Journal of Concrete Structures and Materials (Vol.10, No.3, September 2016) | 303

the beams were defined as 18 NonlinearBeamColumn ele-ments each. The method of the cross section discretization ofthe two flanges is that forty equivalent rectangular Nonlin-earBeamColumn elements are meshed to model each flange,as shown in Fig. 14. The web of the beam has been dividedinto eighteen PCPlaneStress quadrilateral elements. Thesteel ratio in the vertical direction was taken as a very smallnumber to avoid numerical problems in analysis. The ten-dons and concrete were defined by using TendonL01 and

ConcreteR01 constitutive modules in the PCPlaneStresselements, respectively.

3.3 Comparison of Experimental and AnalyticalResults3.3.1 Web-Shear FailureThe analytical and tested load–displacement curves for all

PSFC I-beams in web-shear failure mode are shown inFigs. 15 to 18. These figures show that the results obtainedfrom the numerical analysis correspond well with resultsobtained from the test. The general trend observed in thebeam (R1 to R4) tests, i.e. increase in load carrying capacityof beams with increase in fiber-factor (FF), was accuratelypredicted by the analysis program.Unlike Beams R1, R2 and R3 the prediction of the post-

cracking stiffness of Beam R4 did not match well with theexperimental results. This is due to the fact that while testingBeam R4, the data acquisition system (DAS) got someproblem in the electronic part and recorded invalid valuesand stopped after some time. After resolving the problemwith DAS, the test has been restarted. This invalid data is thereason for the lower cracking value and post cracking stiff-ness observed during the testing of the beam than thoseobtained from the analysis.

Fig. 16 Comparison of analytical and experimental results ofbeam R2.

Fig. 17 Comparison of analytical and experimental results ofbeam R3.

Fig. 15 Comparison of analytical and experimental results ofbeam R1.

Fig. 18 Comparison of analytical and experimental results ofbeam R4.

Fig. 19 Comparison of analytical and experimental results ofbeam RB1.

304 | International Journal of Concrete Structures and Materials (Vol.10, No.3, September 2016)

All the box-beams encountered a local flexural failure attop flange and end block-out of the beams, which pre-vented the accomplishment of expected ultimate shearfailure and anticipated ductility levels (Tadepalli et al.2014). This can be clearly observed from all the curvesshown in Figs. 19 to 22. The analysis was able to satis-factorily predict the ultimate load capacity and ductility ofthe box-beams tested.

3.3.2 Flexure-Shear FailureThe measured and calculated load-deformation curves for

all beams with flexure-shear failure are shown in Figs. 23 to26. Figures 23 and 24 indicate the results from I-beams,while Figs. 25 and 26 show those for box-beams. Thesefigures show that the results obtained from the numericalanalysis correspond well with results obtained from the test.The analysis could well predict the lower load carrying

Fig. 21 Comparison of analytical and experimental results ofbeam RB4.

Fig. 20 Comparison of analytical and experimental results ofbeam RB2.

Fig. 22 Comparison of analytical and experimental results ofbeam RB6.

Fig. 23 Comparison of analytical and experimental results ofbeam R5.

Fig. 24 Comparison of analytical and experimental results ofbeam R6.

Fig. 25 Comparison of analytical and experimental results ofbeam RB3.

International Journal of Concrete Structures and Materials (Vol.10, No.3, September 2016) | 305

capacity of beams tested under flexural shear in comparisonto the beams tested under web shear failure having sameamount of steel fibers. The higher ductility observed in thebehavior of beams under flexural-shear failure, than all theweb shear specimens was also well predicted in the analysis.All the box-beams had a local failure, which prevented

them to reach up to their ultimate shear capacity and also theductility. This can be clearly observed from all the curvesshown in Figs. 25 and 26. The analysis was able to predictthe true ultimate load capacity and ductility.

4. Conclusion

Using the constitutive laws of PSFC established previ-ously, an analytical model was developed and implementedin a FEM program framework (OpenSees) to simulate theshear behavior of the PSFC beams. Using this computerprogram, the load–deflection curves of all the I- and box-beams are simulated with acceptable accuracy.

Acknowledgments

The authors would like to express their gratitude to thefollowing sponsors for this project. They are the State KeyLaboratory for Disaster Reduction of Civil Engineering(Project No. SLDRCE11-MB-01), Tongji University, Chinaand the National Natural Science Foundation of China(Grant No. 51178354). The materials presented are theresearch findings by the authors, and are not necessarilyexpressed for the funding agencies’ opinion.

Open Access

This article is distributed under the terms of the CreativeCommons Attribution 4.0 International License(http://creativecommons.org/licenses/by/4.0/), which per-mits unrestricted use, distribution, and reproduction in anymedium, provided you give appropriate credit to the original

author(s) and the source, provide a link to the CreativeCommons license, and indicate if changes were made.

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Fig. 26 Comparison of analytical and experimental results ofbeam RB5.

306 | International Journal of Concrete Structures and Materials (Vol.10, No.3, September 2016)